Finite Group Databasebeta

← #31792606

Subgroups — #31792606

Order 17536.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578111C211yesmaximal, characteristic, complemented
2D4×C22⋊D137C211yesmaximal, characteristic
3#22577748C211yesmaximal, characteristic, complemented
4#22577722C211yesmaximal, characteristic, complemented
5#22577720C211yesmaximal, characteristic, complemented
6#22577691C211yesmaximal, characteristic, complemented
7#22577634C211yesmaximal, characteristic, complemented
8#22577629C211yesmaximal, characteristic, complemented
9#22577601C211yesmaximal, characteristic, complemented
10#22577552C211yesmaximal, characteristic, complemented
11SD16×Dic137C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13C2×#17984009C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#450×C137C211yesmaximal, characteristic, complemented
16C22×C22⋊D137—21no—
17C2×D4×D137C2211yescharacteristic
18C2×#14555497C2211yescharacteristic, complemented
19C22×#11764040C2211yescharacteristic, complemented
20C2×C8×D137C2211yescharacteristic, complemented
21#17984087C2211yescharacteristic, complemented
22#17984084C2211yescharacteristic
23#17984078—21no—
24#17984068—21no—
25#17984065—21no—
26#17984064C2211yescharacteristic, complemented
27#17984062C2211yescharacteristic
28#17984060—21no—
29#17984059C2211yescharacteristic
30#17984057—21no—
31#17984056—21no—
32Q8×Dic137C2211yescharacteristic, complemented
33D4×Dic137C2211yescharacteristic
34C4×C22⋊D137C2211yescharacteristic
35#17984048—21no—
36C2×C4⋊Dic137C2211yescharacteristic, complemented
37C4⋊C4×D137C2211yescharacteristic, complemented
38#17984020C2211yescharacteristic, complemented
39#17984020C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984012C2211yescharacteristic, complemented
43#17984010C2211yescharacteristic, complemented
44#17984010C2211yescharacteristic, complemented
45#17984009C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984009C2211yescharacteristic, complemented
48#17984009C2211yescharacteristic, complemented
49#17984008C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C22×D4×C137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic
54C2×SD16×C137C2211yescharacteristic, complemented
55C22⋊Q8×C137C2211yescharacteristic, complemented
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60C2×Dic274—21no—
61C2×D548C2311yescharacteristic
62C2×C22⋊D137—21no—
63C2×C22⋊D137—21no—
64C2×C22⋊D137—21no—
65C2×C22⋊D137—21no—
66C2×C22⋊D137—21no—
67C2×C22⋊D137—44no—
68C2×C22⋊D137D411yescharacteristic
69D4×D137—22no—
70D4×D137—41no—
71C22×Dic137—21no—
72C22×Dic137—21no—
73C2×C4×D137—21no—
74C2×C4×D137C2311yescharacteristic
75#14555497—41no—
76C2×#11764040—41no—
77C2×#11764040C2311yescharacteristic
78C2×#11764040C2311yescharacteristic
79C8×D137—41no—
80C4⋊Dic137—21no—
81C4⋊Dic137—21no—
82C4⋊Dic137C2311yescharacteristic
83C4⋊Dic137C2311yescharacteristic
84C4⋊Dic137C2311yescharacteristic
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88#14555489D411yescharacteristic
89C22⋊Dic137—21no—
90C22⋊Dic137—21no—
91C22⋊Dic137D411yescharacteristic
92#14555487—21no—
93#14555487—21no—
94#14555487D411yescharacteristic
95C4×Dic137—21no—
96C4×Dic137C2311yescharacteristic
97C24×C137—21no—
98C2×Q8×C137C2311yescharacteristic
99C2×D4×C137—22no—
100C2×D4×C137—21no—
101C2×D4×C137—41no—
102C2×D4×C137C2311yescharacteristic
103C2×D4×C137C2311yescharacteristic
104C22×C4×C137C2311yescharacteristic
105SD16×C137—22no—
106SD16×C137—41no—
107C2×C8×C137C2311yescharacteristic
108C2×C8×C137C2311yescharacteristic
109C4⋊C4×C137—21no—
110C4⋊C4×C137C2311yescharacteristic
111C22⋊C4×C137—21no—
112C22×D137—21no—
113C22×D137—42no—
114C22×D137—42no—
115C22×D137—81no—
116C22×D137C2×D411yescharacteristic
117Dic274—81no—
118D548—21no—
119D548—21no—
120C22⋊D137—42no—
121C22⋊D137—42no—
122C22⋊D137—42no—
123C22⋊D137—42no—
124C22⋊D137—81no—
125C22⋊D137SD1614yescomplemented
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137—21no—
129C2×Dic137—42no—
130C2×Dic137—81no—
131C2×Dic137C2×D411yescharacteristic
132C2×Dic137C2×D411yescharacteristic
133C2×Dic137C2×D411yescharacteristic
134C4×D137—21no—
135C4×D137—21no—
136C4×D137—81no—
137#11764040—21no—
138#11764040—21no—
139C23×C137—21no—
140C23×C137—21no—
141C23×C137—42no—
142C23×C137—42no—
143C23×C137—81no—
144C23×C137C2×D411yescharacteristic
145Q8×C137—21no—
146Q8×C137—21no—
147D4×C137—21no—
148D4×C137—21no—
149D4×C137—21no—
150D4×C137—42no—
151D4×C137#4112yes—
152C2×C4×C137—21no—
153C2×C4×C137—21no—
154C2×C4×C137—21no—
155C2×C4×C137—21no—
156C2×C4×C137C2×D411yescharacteristic
157C2×C4×C137C2411yescharacteristic
158C8×C137—21no—
159C8×C137—21no—
160C22×C137—21no—
161C22×C137—21no—
162C22×C137—22no—
163C22×C137—41no—
164C22×C137—41no—
165C22×C137—41no—
166C22×C137—41no—
167C22×C137—41no—
168C22×C137—84no—
169C22×C137C2×SD1612yes—
170C22×C137C22×D411yescharacteristic
171C2×D137—21no—
172C2×D137—21no—
173C2×D137—41no—
174C2×D137—41no—
175C2×D137—41no—
176C2×D137—84no—
177C2×D137C2×SD1612yes—
178C4×C137—21no—
179C4×C137—41no—
180C4×C137—41no—
181C4×C137C2×#4111yescharacteristic
182C4×C137H(5,𝔽2)11yescharacteristic
183Dic137—21no—
184Dic137—21no—
185Dic137—41no—
186Dic137—41no—
187Dic137—41no—
188Dic137C2×SD1612yes—
189C2×C137—22no—
190C2×C137—42no—
191C2×C137—81no—
192C2×C137#54611yescharacteristic
193C2×C137C22×SD1611yescharacteristic
194C2×C137#57611yescharacteristic
195D137—22no—
196D137—81no—
197C137#329511yescharacteristic, complemented, Sylow
198#3295—1371nomaximal, Sylow
199#554—1371no—
200D42—1371no—
201#486—1371no—
202#481—1371no—
203#481—1371no—
204#476—1371no—
205#465—1371no—
206#465—1371no—
207#460—1371no—
208#450—1371no—
209#450D13711yescharacteristic, complemented
210C4×SD16—1371no—
211C8×D4—1371no—
212C2×#102—1371no—
213C2×#102—1371no—
214C22×D4—2741no—
215C22×D4—1371no—
216C22×D4C2×D13711yescharacteristic
217C2×SD16—1371no—
218C2×SD16C2×D13711yescharacteristic
219C2×SD16C2×D13711yescharacteristic, complemented
220C22×C8—1371no—
221C22×C8—1371no—
222C4⋊Q8—1371no—
223#127—1371no—
224#125—2741no—
225C22⋊Q8—2741no—
226C22⋊Q8—2741no—
227C22⋊Q8—1371no—
228C22⋊Q8C2×D13711yescharacteristic, complemented
229#121—2741no—
230#121—1371no—
231#121—1371no—
232#120—2741no—
233#120—2741no—
234C4×Q8—1371no—
235C4×D4—2741no—
236C4×D4—1371no—
237C4×D4—1371no—
238C2×C4⋊C4—1371no—
239C2×C4⋊C4—1371no—
240#106—1371no—
241#106—1371no—
242#106—1371no—
243C4⋊C8—1371no—
244#103—1371no—
245#102—1371no—
246#102—1371no—
247#102—1371no—
248#102—1371no—
249#102—1371no—
250#102—1371no—
251#102—1371no—
252#102C2×D13711yescharacteristic
253#102C2×D13711yescharacteristic, complemented
254C22⋊C8—1371no—
255C22⋊C8C2×D13711yescharacteristic, complemented
256C4×C8—1371no—
257C24—21no—
258C24—2741no—
259C2×Q8—2741no—
260C2×Q8C22×D13711yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—2742no—
265C2×D4—2741no—
266C2×D4—2741no—
267C2×D4—2741no—
268C2×D4—2741no—
269C2×D4—2741no—
270C2×D4—5481no—
271C2×D4—5484no—
272C2×D4—1371no—
273C2×D4—1371no—
274C2×D4C22×D13711yescharacteristic
275C2×D4C22×D13711yescharacteristic
276C22×C4—2741no—
277C22×C4—2741no—
278C22×C4—2741no—
279C22×C4—1371no—
280C22×C4C22×D13711yescharacteristic
281SD16—22no—
282SD16—41no—
283SD16—5481no—
284C2×C8—5481no—
285C2×C8—5481no—
286C2×C8—1371no—
287C2×C8—1371no—
288C2×C8C22×D13711yescharacteristic
289C2×C8C22×D13711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—2741no—
292C4⋊C4—2741no—
293C4⋊C4—2741no—
294C4⋊C4—2741no—
295C4⋊C4—2741no—
296C4⋊C4—1371no—
297C4⋊C4—1371no—
298C4⋊C4—1371no—
299C4⋊C4—1371no—
300C4⋊C4C22×D13711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—2741no—
303C22⋊C4—2741no—
304C22⋊C4—2741no—
305C22⋊C4—2741no—
306C22⋊C4—1371no—
307C22⋊C4—1371no—
308C42—2741no—
309C42—1371no—
310C23—21no—
311C23—21no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—2741no—
316C23—5482no—
317C23—5482no—
318C23—10961no—
319C23—1371no—
320C23D4×D13711yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—10961no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—2741no—
329D4—2741no—
330D4—5482no—
331D4—5482no—
332D4—5482no—
333D4—5482no—
334D4—10961no—
335D4—1374no—
336D4#1455550912yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—2741no—
342C2×C4—2741no—
343C2×C4—2741no—
344C2×C4—2741no—
345C2×C4—2741no—
346C2×C4—5482no—
347C2×C4—10961no—
348C2×C4—10961no—
349C2×C4—1371no—
350C2×C4—1371no—
351C2×C4—1371no—
352C2×C4D4×D13711yescharacteristic
353C2×C4C23×D13711yescharacteristic
354C8—21no—
355C8—21no—
356C8—2741no—
357C8—2741no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—2741no—
368C22—2741no—
369C22—5481no—
370C22—5481no—
371C22—5481no—
372C22—10964no—
373C22—1372no—
374C22SD16×D13712yes—
375C22C2×D4×D13711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—2741no—
380C4—2741no—
381C4—5481no—
382C4—5481no—
383C4—5481no—
384C4—1372no—
385C4C2×#1455550911yescharacteristic
386C4#1798413311yescharacteristic
387C2—22no—
388C2—42no—
389C2—81no—
390C2—2742no—
391C2—10961no—
392C2#2257804011yescharacteristic
393C2C2×SD16×D13711yescharacteristic
394C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#450×C137C2
socleC22×C137C22×D4
prime radicalD4×C22⋊D137C2
hypercenter#450D137
hypocenterC137#3295
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22578111C2semidirect
3. #22577748C2semidirect
4. #22577722C2semidirect
5. #22577720C2semidirect
6. #22577691C2semidirect
7. #22577634C2semidirect
8. #22577629C2semidirect
9. #22577601C2semidirect
10. #22577552C2semidirect
11. SD16×Dic137C2semidirect
12. #22577466C2semidirect
13. C2×#17984009C2semidirect
14. #102×D137C2semidirect
15. #450×C137C2semidirect
18. C2×#14555497C22semidirect
19. C22×#11764040C22semidirect
20. C2×C8×D137C22semidirect
21. #17984087C22semidirect
26. #17984064C22semidirect
32. Q8×Dic137C22semidirect
36. C2×C4⋊Dic137C22semidirect
37. C4⋊C4×D137C22semidirect
38. #17984020C22semidirect
39. #17984020C22semidirect
40. #17984019C22semidirect
41. #17984017C22semidirect
42. #17984012C22semidirect
43. #17984010C22semidirect
44. #17984010C22semidirect
47. #17984009C22semidirect
48. #17984009C22semidirect
49. #17984008C22semidirect
50. #17984000C22semidirect
51. C8×Dic137C22semidirect
54. C2×SD16×C137C22semidirect
55. C22⋊Q8×C137C22semidirect
57. #102×C137C22semidirect
58. C22⋊C8×C137C22semidirect
125. C22⋊D137SD16semidirect
197. C137#3295semidirect
209. #450D137semidirect
219. C2×SD16C2×D137semidirect
228. C22⋊Q8C2×D137semidirect
253. #102C2×D137semidirect
255. C22⋊C8C2×D137semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225781112
D4×C22⋊D1372
#225777482
#225777222
#225777202
#225776912
#225776342
#225776292
#225776012
#225775522
SD16×Dic1372
#225774662
C2×#179840092
#102×D1372
#450×C1372
#3295137